scholarly journals Mean Flow Effects in the Faraday Instability

2003 ◽  
Vol 17 (22n24) ◽  
pp. 4278-4283
Author(s):  
Elena Martín ◽  
Carlos Martel ◽  
José M. Vega
Keyword(s):  
Surface Waves ◽  
Navier Stokes ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Viscosity Limit ◽  
Faraday Instability ◽  
Wave Patterns ◽  
Long Time ◽  
Flow Effects ◽  
The Stability

We study the weakly nonlinear evolution of Faraday waves in a 2D container that is vertically vibrated. In the small viscosity limit, the evolution of the surface waves is coupled to a non-oscillatory mean flow that develops in the bulk of the container. The corresponding long time (Navier-Stokes+amplitude) equations are derived and analyzed numerically. The results indicate that the (usually ignored) mean flow plays an essential role in the stability of the surface waves and in the bifurcated wave patterns.

scholarly journals Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Mohammed Rizwan Sadiq Iqbal
Keyword(s):  
Magnetic Field ◽  
Inertial Force ◽  
Transverse Magnetic ◽  
Nonlinear Analyses ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Strength ◽  
Long Time ◽  
The Stability ◽  
The One

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.

scholarly journals Eddy-Driven Recirculations from a Localized Transient Forcing

2012 ◽  
Vol 42 (3) ◽  
pp. 430-447 ◽  
Author(s):  
Stephanie Waterman ◽  
Steven R. Jayne
Keyword(s):  
Group Velocity ◽  
Rossby Wave ◽  
Wave Spectrum ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Recirculation Gyres ◽  
Flow Interactions ◽  
Flow Effects ◽  
Wave Limit ◽  
Many Sources

Abstract The generation of time-mean recirculation gyres from the nonlinear rectification of an oscillatory, spatially localized vorticity forcing is examined analytically and numerically. Insights into the rectification mechanism are presented and the influence of the variations of forcing parameters, stratification, and mean background flow are explored. This exploration shows that the efficiency of the rectification depends on the properties of the energy radiation from the forcing, which in turn depends on the waves that participate in the rectification process. The particular waves are selected by the relation of the forcing parameters to the available free Rossby wave spectrum. An enhanced response is achieved if the parameters are such to select meridionally propagating waves, and a resonant response results if the forcing selects the Rossby wave with zero zonal group velocity and maximum meridional group velocity, which is optimal for producing rectified flows. Although formulated in a weakly nonlinear wave limit, simulations in a more realistic turbulent system suggest that this understanding of the mechanism remains useful in a strongly nonlinear regime with consideration of mean flow effects and wave–mean flow interaction now needing to be taken into account. The problem presented here is idealized but has general application in the understanding of eddy–eddy and eddy–mean flow interactions as the contrasting limit to that of spatially broad (basinwide) forcing and is relevant given that many sources of oceanic eddies are localized in space.

Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves

1983 ◽  
Vol 133 ◽  
pp. 113-132 ◽  
Author(s):  
Peter A. E. M. Janssen
Keyword(s):  
Wave Spectrum ◽  
Inhomogeneous Field ◽  
Time Behaviour ◽  
Phase Mixing ◽  
Weakly Nonlinear ◽  
Large Bandwidth ◽  
Long Time Behaviour ◽  
Homogeneous Wave ◽  
Long Time ◽  
The Stability

In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width σ of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth – the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi–Pasta–Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum.

The onset of meandering in a barotropic jet

2001 ◽  
Vol 449 ◽  
pp. 85-114 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO
Keyword(s):  
Nonlinear Theory ◽  
Stability Boundary ◽  
Wave Model ◽  
Inviscid Limit ◽  
Reduced Model ◽  
Mean Flow ◽  
Single Wave ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

This study explores the dynamics of an unstable jet of two-dimensional, incompressible fluid on the beta-plane. In the inviscid limit, standard weakly nonlinear theory fails to give a low-order description of this problem, partly because the simple shape of the unstable normal mode is insufficient to capture the structure of the forming pattern. That pattern takes the form of ‘cat's eyes’ in the vorticity distribution which develop inside the modal critical layers (slender regions to either side of the jet's axis surrounding the levels where the modal wave speed matches the mean flow). Asymptotic expansions furnish a reduced model which is a version of what is known as the single-wave model in plasma physics. The reduced model predicts that the amplitude of the unstable mode saturates at a relatively low level and is not steady. Rather, the amplitude evolves aperiodically about the saturation level, a result with implications for Lagrangian transport theories. The aperiodic amplitude ‘bounces’ are intimately connected with sporadic deformations of the vortices within the cat's eyes. The theory is compared with numerical simulations of the original governing equations. Slightly asymmetrical jets are also studied. In this case the neutral modes along the stability boundary become singular; an extension of the weakly nonlinear theory is presented for these modes.

scholarly journals The impedance boundary condition for acoustics in swirling ducted flow

2018 ◽  
Vol 849 ◽  
pp. 645-675 ◽  
Author(s):  
Vianney Masson ◽  
James R. Mathews ◽  
Stéphane Moreau ◽  
Hélène Posson ◽  
Edward J. Brambley
Keyword(s):  
Boundary Condition ◽  
Surface Waves ◽  
Swirling Flow ◽  
Swirling Flows ◽  
Impedance Boundary Condition ◽  
Mean Flow ◽  
Thin Boundary Layer ◽  
Impedance Boundary ◽  
Unstable Surface ◽  
The Stability

The acoustics of a straight annular lined duct containing a swirling mean flow is considered. The classical Ingard–Myers impedance boundary condition is shown not to be correct for swirling flow. By considering behaviour within the thin boundary layers at the duct walls, the correct impedance boundary condition for an infinitely thin boundary layer with swirl is derived, which reduces to the Ingard–Myers condition when the swirl is set to zero. The correct boundary condition contains a spring-like term due to centrifugal acceleration at the walls, and consequently has a different sign at the inner (hub) and outer (tip) walls. Examples are given for mean flows relevant to the interstage region of aeroengines. Surface waves in swirling flows are also considered, and are shown to obey a more complicated dispersion relation than for non-swirling flows. The stability of the surface waves is also investigated, and as in the non-swirling case, one unstable surface wave per wall is found.

Stabilizing the long-time behavior of the forced Navier–Stokes and damped Euler systems by large mean flow

2018 ◽  
Vol 369 ◽  
pp. 18-29 ◽  
Author(s):  
Jacek Cyranka ◽  
Piotr B. Mucha ◽  
Edriss S. Titi ◽  
Piotr Zgliczyński
Keyword(s):  
Navier Stokes ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Mean Flow ◽  
Euler Systems ◽  
Long Time

Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow

2008 ◽  
Vol 602 ◽  
pp. 403-426 ◽  
Author(s):  
E. J. BRAMBLEY ◽  
N. PEAKE
Keyword(s):  
Surface Waves ◽  
Acoustic Waves ◽  
Thin Shell ◽  
Sudden Change ◽  
Acoustic Scattering ◽  
Rigid Wall ◽  
Instability Wave ◽  
Mean Flow ◽  
The Stability ◽  
Mass Spring

We consider the stability of small perturbations to a uniform inviscid compressible flow within a cylindrical linear-elastic thin shell. The thin shell is modelled using Flügge's equations, and is forced from the inside by the fluid, and from the outside by damping and spring forces. In addition to acoustic waves within the fluid, the system supports surface waves, which are strongly coupled to the thin shell. Stability is analysed using the Briggs–Bers criterion, and the system is found to be either stable or absolutely unstable, with absolute instability occurring for sufficiently small shell thicknesses. This is significantly different from the stability of a thin shell containing incompressible fluid, even for parameters for which the fluid would otherwise be expected to behave incompressibly (for example, water within a steel thin shell). Asymptotic expressions are derived for the shell thickness separating stable and unstable behaviour.We then consider the scattering of waves by a sudden change in the duct boundary from rigid to thin shell, using the Wiener–Hopf technique. For the scattering of an inbound acoustic wave in the rigid-wall section, the surface waves are found to play an important role close to the sudden boundary change. The solution is given analytically as a sum of duct modes.The results in this paper add to the understanding of the stability of surface waves in models of acoustic linings in aeroengine ducts. The oft-used mass–spring–damper model is regularized by the shell bending terms, and even when these terms are very small, the stability and scattering results are quite different from what has been claimed for the mass–spring–damper model. The scattering results derived here are exact, unique and causal, without the need to apply a Kutta-like condition or to include an instability wave. A movie is available with the online version of the paper.

scholarly journals Modulated surface waves in large-aspect-ratio horizontally vibrated containers

2007 ◽  
Vol 579 ◽  
pp. 271-304 ◽  
Author(s):  
FERNANDO VARAS ◽  
JOSÉ M. VEGA
Keyword(s):  
Aspect Ratio ◽  
Surface Waves ◽  
Linear Analysis ◽  
Harmonic Waves ◽  
Large Aspect Ratio ◽  
Mean Flow ◽  
Horizontal Vibration ◽  
Weakly Nonlinear ◽  
Qualitative And Quantitative ◽  
Threshold Amplitude

We consider the harmonic and subharmonic modulated surface waves that appear upon horizontal vibration along the surface of the liquid in a two-dimensional large-aspect-ratio (length large compared to depth) container, whose depth is large compared to the wavelength of the surface waves. The analysis requires us also to consider an oscillatory bulk flow and a viscous mean flow. A weakly nonlinear description of the harmonic waves is made which provides the threshold forcing amplitude to trigger harmonic instabilities, which are of various qualitatively different kinds. A linear analysis provides the threshold amplitude for the appearance of subharmonic waves through a subharmonic instability. The results obtained are used to make several specific qualitative and quantitative predictions.

scholarly journals Linear stability and weakly nonlinear analysis of the flow past rotating spheres

2016 ◽  
Vol 807 ◽  
pp. 62-86 ◽  
Author(s):  
V. Citro ◽  
J. Tchoufag ◽  
D. Fabre ◽  
F. Giannetti ◽  
P. Luchini
Keyword(s):  
Three Dimensional ◽  
Base Flow ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Weakly Nonlinear ◽  
Rotating Sphere ◽  
Flow Past ◽  
The Stability ◽  
Planar Symmetry ◽  
Flow Past A Sphere

We study the flow past a sphere rotating in the transverse direction with respect to the incoming uniform flow, and particularly consider the stability features of the wake as a function of the Reynolds number $Re$ and the sphere dimensionless rotation rate $\unicode[STIX]{x1D6FA}$. Direct numerical simulations and three-dimensional global stability analyses are performed in the ranges $150\leqslant \mathit{Re}\leqslant 300$ and $0\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.2$. We first describe the base flow, computed as the steady solution of the Navier–Stokes equation, with special attention to the structure of the recirculating region and to the lift force exerted on the sphere. The stability analysis of this base flow shows the existence of two different unstable modes, which occur in different regions of the $Re/\unicode[STIX]{x1D6FA}$ parameter plane. Mode I, which exists for weak rotations ($\unicode[STIX]{x1D6FA}<0.4$), is similar to the unsteady mode existing for a non-rotating sphere. Mode II, which exists for larger rotations ($\unicode[STIX]{x1D6FA}>0.7$), is characterized by a larger frequency. Both modes preserve the planar symmetry of the base flow. We detail the structure of these eigenmodes, as well as their structural sensitivity, using adjoint methods. Considering small rotations, we then compare the numerical results with those obtained using weakly nonlinear approaches. We show that the steady bifurcation occurring for $Re>212$ for a non-rotating sphere is changed into an imperfect bifurcation, unveiling the existence of two other base-flow solutions which are always unstable.

scholarly journals Weak compressibility of surface wave turbulence

2007 ◽  
Vol 593 ◽  
pp. 281-296 ◽  
Author(s):  
MARIJA VUCELJA ◽  
ITZHAK FOUXON
Keyword(s):  
Surface Waves ◽  
Small Amplitude ◽  
Density Fluctuations ◽  
Wave Turbulence ◽  
Small Scale ◽  
Weakly Nonlinear ◽  
Weakly Compressible ◽  
Typical Time ◽  
Long Time ◽  
Finite Exponent

We study the growth of small-scale inhomogeneities in the density of particles floating in weakly nonlinear small-amplitude surface waves. Despite the small amplitude, the accumulated effect of the long-time evolution may produce a strongly inhomogeneous distribution of the floaters: density fluctuations grow exponentially with a small but finite exponent. We show that the exponent is of sixth or higher order in wave amplitude. As a result, the inhomogeneities do not form within typical time scales of the natural environment. We conclude that the turbulence of surface waves is weakly compressible and alone it cannot be a realistic mechanism of the clustering of matter on liquid surfaces.

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